Is it permissible to start the process on Shabbat, or if not, to start it before, and let the process run into Shabbat? This question concerns the actual soaking; the straining typically done to such a brew is a separate issue.

I would reframe the question in a different way. French press has a huge problem of the strainer, as you have already noted. Why not just inquire about the soaking process and leave out the straining or the particular method (French press) out of question?
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Seth JNov 20 '12 at 16:31

2

I explicitly excluded straining from the question. Based on your feedback, I removed reference to the press.
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yitznewtonNov 20 '12 at 16:58

(Just for the record both the Aruch haShulchan and the Mishnah Berurah prohibit brewing tea even in a kli sheni on Shabbat. Others argue, but I wouldn't present it as a clear-cut rule.)
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Double AA♦Nov 20 '12 at 20:05

2 Answers
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It is a matter of extensive debate. The opinion of the Kitzur Shulchan Aruch cited by @Danny Schoemann is agreed upon by the Aruch Hashulchan (OC 318:29). However, the Taz (318:5) infers from the Rema that simply making something edible is not halachically considered cooking when heat is not involved. The Mishna Berura (318:37) rules in accordance with the Taz.

Food that is impossible to eat without first being rinsed, may not be
rinsed on Shabbat even in cold water.

Salty fish may be rinsed in cold water, since it was (somewhat) edible
even before being rinsed.

Since soaking grounds in cold water for an extended period makes them edible, it should be forbidden to do so on Shabbat.

Starting to do so before Shabbat should be OK; it's no different than starting your cholent before Shabbat. (Actually may be easier than that, since there's zero chance of trying to improve the results by playing with the flame.)

I don't necessarily agree with "soaking grounds in cold water for an extended period makes them edible": you don't eat the grounds.
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msh210♦Nov 21 '12 at 14:33

Also, if this is the case why would anyone permit tea? You can soak tea leaves in cold water and make tea - yet even hot water is permitted! (By some, I don't do it personally.)
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ArielNov 21 '12 at 22:19